Solving Cubic Equations Arithmetic Progression Roots Find M
Hey guys! Today, we're diving deep into a fascinating math problem straight from Colegio Ciencias y Humanidades 5. This isn't just any equation; it involves roots in arithmetic progression, which adds a layer of complexity that makes it super interesting. So, buckle up, math enthusiasts, as we dissect this problem step-by-step and uncover the value of 'm'.
The Challenge: Unraveling the Cubic Equation
Our mission, should we choose to accept it, is to solve the following cubic equation:
-x³ + 15x² + mx - 45 = 0
The twist? We know that the roots of this equation are in arithmetic progression. This little nugget of information is our key to unlocking the value of 'm'. But before we jump into the solution, let's break down what arithmetic progression means and how it applies to the roots of a cubic equation.
Arithmetic Progression: The Foundation of Our Solution
Arithmetic progression (AP), in simple terms, is a sequence of numbers where the difference between any two consecutive terms is constant. Think of it like climbing stairs where each step is the same height. This constant difference is known as the common difference.
For example, the sequence 2, 5, 8, 11... is an arithmetic progression with a common difference of 3. Each term is obtained by adding 3 to the previous term. Understanding this concept is crucial because the roots of our equation follow this pattern.
Roots of a Cubic Equation: A Quick Recap
A cubic equation, like the one we're dealing with, has three roots. These roots are the values of 'x' that make the equation equal to zero. Now, if these roots are in arithmetic progression, we can represent them in a specific way that simplifies our calculations. This is where the magic begins!
The Strategy: Representing Roots in Arithmetic Progression
Here's the clever trick: When dealing with roots in arithmetic progression, we can represent them as follows:
- a - d
- a
- a + d
Where:
- 'a' is the middle term (which is also the average of the three terms).
- 'd' is the common difference.
This representation is super handy because it introduces symmetry into our problem. When we use Vieta's formulas (which we'll discuss shortly), some terms will conveniently cancel out, making our calculations much easier.
Vieta's Formulas: Our Secret Weapon
Vieta's formulas are a set of relationships between the coefficients of a polynomial and its roots. For a cubic equation of the form:
ax³ + bx² + cx + d = 0
Vieta's formulas tell us:
- Sum of roots: -b/a
- Sum of pairwise products of roots: c/a
- Product of roots: -d/a
These formulas are our secret weapon in solving this problem. They allow us to relate the roots (which we've represented in terms of 'a' and 'd') to the coefficients of our equation.
The Execution: Solving for 'm' Step-by-Step
Now, let's put our strategy into action. Our equation is:
-x³ + 15x² + mx - 45 = 0
Comparing this to the general form ax³ + bx² + cx + d = 0, we have:
- a = -1
- b = 15
- c = m
- d = -45
And our roots are (a - d), a, and (a + d).
Step 1: Sum of the Roots
Using Vieta's first formula, the sum of the roots is -b/a. So,
(a - d) + a + (a + d) = -15 / -1
Notice how the 'd' terms cancel out beautifully, leaving us with:
3a = 15
Solving for 'a', we get:
a = 5
This is a major breakthrough! We've found the middle root of our equation.
Step 2: Product of the Roots
Next, let's use Vieta's third formula, which relates the product of the roots to the coefficients. The product of the roots is -d/a. Therefore,
(a - d) * a * (a + d) = -(-45) / -1
Simplifying,
(a - d) * a * (a + d) = -45
We already know that a = 5, so let's substitute that in:
(5 - d) * 5 * (5 + d) = -45
Divide both sides by 5:
(5 - d) * (5 + d) = -9
This looks like a difference of squares! Expanding the left side, we get:
25 - d² = -9
Rearranging, we have:
d² = 34
So,
d = ±√34
Step 3: Sum of Pairwise Products of Roots
Now, for the grand finale, we'll use Vieta's second formula, which deals with the sum of the pairwise products of the roots. This is where 'm' comes into play. The formula states that the sum of pairwise products is c/a. So,
(a - d) * a + a * (a + d) + (a - d) * (a + d) = m / -1
Let's expand and simplify. Remember, a = 5 and d² = 34:
(5 - d) * 5 + 5 * (5 + d) + (5 - d) * (5 + d) = -m
25 - 5d + 25 + 5d + 25 - d² = -m
The '5d' and '-5d' terms cancel out, leaving us with:
75 - d² = -m
Substitute d² = 34:
75 - 34 = -m
41 = -m
Finally, we've cracked the code:
m = -41
The Verdict: m = -41
And there you have it! The value of 'm' in the equation -x³ + 15x² + mx - 45 = 0, given that its roots are in arithmetic progression, is -41.
Key Takeaways from Our Math Adventure
This problem was a fantastic exercise in applying several key mathematical concepts:
- Arithmetic Progression: Understanding the pattern and representing the roots accordingly was crucial.
- Vieta's Formulas: These formulas provided the essential link between the roots and the coefficients of the equation.
- Algebraic Manipulation: Skillful simplification and solving of equations were key to arriving at the solution.
Why This Matters: The Power of Problem-Solving
Problems like these aren't just about finding the right answer; they're about honing our problem-solving skills. The ability to break down a complex problem into smaller, manageable steps, identify the relevant concepts, and apply them strategically is a skill that's valuable far beyond the math classroom. So, keep challenging yourselves, keep exploring, and keep those mathematical gears turning!